A right circular solid cylinder has radius of base 7 cm and height 28 cm. It is melted and recast into a cuboid whose side lengths are in the ratio 2 : 3 : 6. Assuming volume is conserved during recasting, what is the total surface area of the cuboid in square centimetres? Take pi = 22/7.

Introduction / Context:
This question combines solid geometry with the idea of recasting solids while conserving volume. A right circular cylinder is melted and formed into a cuboid with given proportional side lengths. Candidates must equate volumes of the original and the new solid, determine the actual dimensions of the cuboid from the ratio, and finally calculate the total surface area of the cuboid. Careful handling of pi and cube roots is important for accuracy.

Given Data / Assumptions:

• Original solid is a right circular cylinder with radius r = 7 cm and height h = 28 cm.
• Volume of the cylinder is taken with pi = 22/7.
• The melted material is recast into a cuboid.
• Sides of the cuboid are in ratio 2 : 3 : 6.
• Volume is conserved during recasting.
• We are asked to find the total surface area of the cuboid.

Concept / Approach:
First compute the volume of the original cylinder using V = pi * r^2 * h. Then express the cuboid dimensions as 2k, 3k, and 6k for some scale factor k. The volume of the cuboid is then Vc = 2k * 3k * 6k = 36k^3. Since volume is conserved, V = Vc, allowing us to solve for k. Once k is known numerically, we can find each side length. Finally, the total surface area of a cuboid with side lengths a, b, c is given by TSA = 2(ab + bc + ca).

Step-by-Step Solution:
Step 1: Volume of the cylinder: V = pi * r^2 * h = (22/7) * 7^2 * 28 = (22/7) * 49 * 28.Step 2: Simplify: (22/7) * 49 = 22 * 7, so V = 22 * 7 * 28 = 4312 cm^3.Step 3: Let cuboid sides be 2k, 3k, and 6k. Then cuboid volume Vc = 2k * 3k * 6k = 36k^3.Step 4: Equate volumes: 36k^3 = 4312, so k^3 = 4312 / 36 = 1078 / 9.Step 5: Numerically, k ≈ (1078 / 9)^(1/3) ≈ 4.93. Hence sides are approximately a = 2k ≈ 9.86 cm, b = 3k ≈ 14.79 cm, and c = 6k ≈ 29.58 cm.Step 6: Compute TSA = 2(ab + bc + ca) ≈ 2(9.86 * 14.79 + 14.79 * 29.58 + 29.58 * 9.86) ≈ 1749.5 cm2.

Verification / Alternative check:
One can check that the approximate sides maintain the given volume. Using the approximate values of a, b, and c, the cuboid volume ab c is very close to 4312 cm3, matching the original cylinder volume. Recomputing TSA with more precise decimal places for k gives a value around 1749.51 cm2, which rounds to 1749.5 cm2. Since all other given options are significantly different in magnitude, this confirms that 1749.5 cm2 is the closest and correct total surface area for the recast cuboid.

Why Other Options Are Wrong:

• 1764 cm2 is larger and would require slightly larger side lengths than allowed by the conserved volume.
• 1728 cm2 is smaller and would require slightly smaller sides, again contradicting the volume conservation.
• 1800 cm2 and 1690 cm2 are farther away from the accurate computation and do not agree with the derived dimensions.

Common Pitfalls:

• Using an incorrect volume formula for the cylinder or forgetting to square the radius.
• Misinterpreting the side ratio 2 : 3 : 6 and not assigning them as 2k, 3k, and 6k.
• Rounding k too early and then getting a noticeably different surface area value.

Final Answer:
The total surface area of the cuboid formed is approximately 1749.5 cm2.